Why is the variance of \( X + Y \) measured perpendicular to \( y = -x \)?
1. Lines of Constant Sum: The equation \( X + Y = C \) (or \( y = -x + C \)) represents lines with a slope of -1. Every point on a specific red line has the same sum.
2. Maximum Change: To change the sum \( C \) as quickly as possible, you must move perpendicular to these lines, which is along the green line \( y = x \).
3. Projection: The variance of \( X + Y \) is the spread of the data when projected onto this perpendicular direction (the green axis).
4. Standard Deviation Triangle: The lengths of these sides represent Standard Deviations (\( \sigma \)). When correlation (\( \rho \)) changes, the sides act like a mechanical hinge, pivoting away from the 90-degree Cartesian axes to mathematically close the new hypotenuse.
5. Coordinates vs. Sums (The \( 1/\sqrt{2} \) Factor): The scatterplot plots coordinates, not mathematical sums. If you draw a diagonal line exactly 2 units long, it physically ends at coordinates \((1.41, 1.41)\)—pointing to an oversized sum of 2.82! To make the line accurately point to dots representing a sum of 2, we must scale the physical drawing down by \( 1/\sqrt{2} \) so it lands perfectly at \((1, 1)\). This translates the "language of sums" into the "language of screen pixels."
Acts as the hinge angle between the std dev vectors.
\( \text{Var}(X+Y) = \sigma_X^2 + \sigma_Y^2 + 2\rho\sigma_X\sigma_Y \)
Value: 2.00